What is the integral of (1+y^2)/(y^3) ?

The integral of (1+y^2)/(y^3) is -1/(2y^2)+ln|y|+C

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integral of (1+y^2)/(y^3)

\int \frac{1 + y ^{2}}{y ^{3}} d y

- \frac{1}{2 y ^{2}} + ln ∣ y ∣ + C

Solution steps

\int \frac{1 + y ^{2}}{y ^{3}} d y

Expand

\frac{1 + y ^{2}}{y ^{3}} : \frac{1}{y ^{3}} + \frac{1}{y}

= \int \frac{1}{y ^{3}} + \frac{1}{y} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{y ^{3}} d y + \int \frac{1}{y} d y

\int \frac{1}{y ^{3}} d y = - \frac{1}{2 y ^{2}}

\int \frac{1}{y} d y = ln ∣ y ∣

= - \frac{1}{2 y ^{2}} + ln ∣ y ∣

Add a constant to the solution

= - \frac{1}{2 y ^{2}} + ln ∣ y ∣ + C

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What is the integral of (1+y^2)/(y^3) ?
The integral of (1+y^2)/(y^3) is -1/(2y^2)+ln|y|+C